• Understand key timeseries concepts and features
• See example timeseries that we'll use on the course
• Understand the types of analysis that we want to do, and why

• A collection of data indexed by (strictly increasing) time
• Can be regular or irregular
• Usually one value per time

“With one dimension marching along to the regular rhythm of seconds, minutes, hours, days, weeks, months, years, centuries, or millennia, the natural ordering of the time scale gives this design a strength and efficiency of interpretation found in no other graphic arrangement.”

Edward R. Tufte The Visual Display of Quantitative Information p. 28

Tenth or eleventh century time series showing the position of the planets with time. http://euclid.psych.yorku.ca/SCS/Gallery/milestone/sec2.html

Diagram showing the distance of the planets to the earth in 1732, also showing a complete lunar eclipse and a partial solar eclipse in that year Nicolaas Kruik 1678 - 1754 Dutch Astronomer & Meteorologist

A graph of solar warming vs. lattitude.
Johann Heinrich Lambert 1728 - 1777

Willaim Playfair's trade-balance time-series chart, published in his Commercial and Political Atlas, 1786

What do we use time series methods for? Often, we are trying to do at least one of the following:

1. Description: what is going on?
2. Understanding: how is it going on, and why?
3. Prediction: What is going to go on next?

• HadCRUT4 Annual globally averaged surface temperature
• How do we separate the structure and the random noise?

• Using a line plot gives a slightly different impression.

• Church and White
• There appears relatively less noise in this data compared to the trend

• This AirPassengers data set shows heteroskedasticity

• This data shows heteroskedacity and regime changes

Monthly NH sea ice Anomaly from 1978 to present.

plot(ldeaths)

#with(ldeaths,plot(Age,d18o,type='l'))

Monthly Central England Temperature (CET) from 1659

• Seasonality [on short term] & trend?

• Trend, seasonality, heteroskedacity?, break points?

D18O? Global financial crash? Regime changes

• Autocorrelation is the correlation of a random process with itself at a different time.
• some created examples here

• In a stationary timeseries mean, variance and autocorrelation structure do not change over time.

• A timeseries of the differences between the values.
• Differencing can be used used to make a non-stationary trend stationary.

• Stationarity
• Lag
• Detrending and differencing
• Seasonality removal
• Smoothing & filtering
• Stochastic processes & Random walks

An essay towards solving a problem on the doctrine of chances (1763)

\[P(A|B) = \frac{P(B|A) P(A)}{P(B)}\]

• Bayesian statistics is based on an interpretation of Bayes' theorem
• All quantities are divided up into data (i.e. things which have been observed) and parameters (i.e. things which haven't been observed)
• We use Bayes' interpretation of the theorem to get the posterior probability distribution, the probability of the unobserved given the observed
• Used now in almost all areas of statistical application (finance, medicine, environmetrics, gambling, etc, etc)

Bayes' theorem can be written in words as:

\[\mbox{posterior is proportional to likelihood times prior}\] … or … \[\mbox{posterior} \propto \mbox{likelihood} \times \mbox{prior}\]

Each of the three terms posterior, likelihood, and prior are probability distributions (pdfs).

In a Bayesian model, every item of interest is either data (which we will write as \(x\)) or parameters (which we will write as \(\theta\)). Often the parameters are divided up into those of interest, and other nuisance parameters

Bayes' equation is usually written mathematically as: \[p(\theta|x) \propto p(x|\theta) \times p(\theta)\] or, more fully: \[p(\theta|x) = \frac{p(x|\theta) \times p(\theta)}{p(x)}\]

• The posterior is the probability of the parameters given the data
• The likelihood is the probability of observing the data given the parameters (unknowns)
• The prior represents external knowledge about the parameters

• An ecologist listens for the calls of the southern brown tree frog (Litoria ewingi)
• She wants to know the mean length of the calls
• We will assume that the standard deviation of the calls is known to be 0.8 seconds
• She hears a call of length 3.1 seconds
• A study conducted the previous year estimated the mean to be 2.3 seconds with standard error 0.5 seconds

If we assume that the calls she hears are normally distributed then \(x\) follows a normal distribution with mean \(\theta\) and standard deviation 0.8s, written \(x|\theta \sim N(\theta,0.8^2)\). The prior distribution is \(\theta \sim N(2.3,0.5^2)\).

Code used to produce previous plot:

# Create grid for theta
theta = seq(0,6,length=100)
# Evalutate prior, likelihood and posterior
prior = dnorm(theta,mean=2.3,sd=0.5)
likelihood = dnorm(3.1,mean=theta,sd=0.8)
posterior = prior*likelihood
# Produce plot
plot(theta,likelihood/sum(likelihood),type='l',
     ylab='Probability',ylim=c(0,0.06))
lines(theta,prior/sum(prior),col='red')
lines(theta,posterior/sum(posterior),col='blue')
legend('topright',legend=c('Likelihood','Prior',
                           'Posterior'),
       col=c('black','red','blue'),lty=1)

The Bayesian approach has numerous advantages:

• It's easier to build complex models and to analyse the parameters you want directly
• We automatically obtain the best parameter estimates and their uncertainty from the posterior samples
• It allows us to get away from (terrible) null hypothesis testing and \(p\)-values

• The Bayesian bible: Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2013). Bayesian Data Analysis, Third Edition. CRC Press.
• The MCMC bible: Brooks, S., Gelman, A., Jones, G., & Meng, X. (2011). Handbook of Markov Chain Monte Carlo. CRC Press.
• Something simpler: McCarthy, M. A. (2007). Bayesian Methods for Ecology. Cambridge University Press.

• Bayesian statistical models involve a likelihood and a prior. These both need to be carefully chosen. From these we create a posterior distribution
• The likelihood represents the information about the data generating process, the prior represents information about the unknown parameters
• We usually create and analyse samples from the posterior probability distribution of the unknowns (the parameters) given the knowns (the data)
• From the posterior distribution we can create means, medians, standard deviations, credible intervals, etc